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Unformatted text preview: 254 4 Interest Rate Derivative Securities V ( r, T + ) V ( r, T ) = Z T + T V t dt = Z T + T ( t T ) dt = 1 . That is, V ( r, T ) = V ( r, T + ) + 1 = 1 . In the first problem V ( r, T ) = V ( r, T + ) is also identically equal to one. Consequently, from T to any t < T , the solutions of the two problems are the same. Actually the second problem can be understood as another form of the first problem. 14. Consider the problem: V s 1 t + 1 2 w 2 2 V s 1 r 2 + ( u w ) V s 1 r rV s 1 + 2 N k =1 ( t T k/ 2) = 0 , r l r r u , t T + N, V s 1 ( r, T + N ) = 0 , r l r r u . Show that V s 1 ( r, T ) gives the sum of values of 2 N zerocoupon bonds with maturities 1 / 2 , 1 , 3 / 2 , , N years. Solution : The solution of the given problem is the sum of the solutions of the fol lowing 2 N problems V s 1 k t + 1 2 w 2 2 V s 1 k r 2 + ( u w ) V s 1 k r rV s 1 k + ( t T k/ 2) = 0 , r l r r u , t T + N, V s 1 k ( r, T + N ) = 0 , r l r r u , k = 1 , 2 , , 2 N. It is clear that V s 1 k ( r, t ) = 0 for t ( T + k/ 2 , T + N ] and for t < T + k/ 2, V s 1 k ( r, t ) satisfies V s 1 k t + 1 2 w 2 2 V s 1 k r 2 + ( u w ) V s 1 k r rV s 1 k + ( t T k/ 2) = 0 , r l r r u , t T + k/ 2 , V s 1 k ( r, T + k/ 2) = 0 , r l r r u . 4 Interest Rate Derivative Securities 255 In Problem 13 we have shown that this problem can be rewritten as...
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This note was uploaded on 02/11/2010 for the course MATH 6203 taught by Professor Zhu during the Spring '10 term at University of North Carolina Wilmington.
 Spring '10
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